THE MULTIPLICATIVE PROPERTY CHARACTERIZES p AND Lp NORMS

THE MULTIPLICATIVE PROPERTY CHARACTERIZES p AND Lp NORMS GUILLAUME AUBRUN AND ION NECHITA Abstract. We show that p norms are characterized as the unique norms which are both invariant under coordinate permutation and multiplicative with respect to tensor products. Similarly, the Lp norms are the unique rearrangement-invariant norms on a probability space such that ?XY ? = ?X?·?Y ? for every pair X,Y of independent random variables. Our proof combines the tensor power trick and Cramer's large deviation theorem. 1. Introduction The p and Lp spaces are among the most important examples of Banach spaces and have been widely investigated (see e.g. [2] for a survey). In this note, we exhibit a characterization of the p/Lp norms by a simple algebraic identity: the multiplicative property. In the case of p norms, this property reads as ?x? y? = ?x? · ?y? for every (finite) sequences x, y. In the case of Lp norms, it becomes ?XY ? = ?X? · ?Y ? whenever X,Y are independent (bounded) random variables. There are many examples of theorems showing how special are p/Lp spaces among Banach spaces. An early axiomatic characterization of p/Lp spaces goes back to Bohnenblust [6]: among Banach lattices, they are the only spaces in which ?x + y? depends only on ?x? and ?y? whenever x, y are orthogonal.

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